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Digital Signal Processing Handbook P60

Much of modern digital signal processing is concernedwith the extraction of information fromsignals whichare noisy, orwhichbehave randomlywhile still revealingsomeattributeor parameterof a system or environment under observation. The term in popular use now for this kind of computation is statistical signal processing, and much of this Handbook is devoted to this very subject. Statistical signal processing is classical statistical inference applied to problems of interest to electrical engineers, with the added twist that answers are often required in “real time”, perhaps seconds or less. Thus, computational algorithms are often studied hand-in-hand with statistics | Fuhrmann D.R. Complex Random Variables and Stochastic Processes Digital Signal Processing Handbook Ed. Vijay K. Madisetti and Douglas B. Williams Boca Raton CRC Press LLC 1999 1999 by CRC Press LLC 60 Complex Random Variables and Stochastic Processes Daniel R. Fuhrmann Washington University 60.1 Introduction 60.2 Complex Envelope Representations of Real Bandpass Stochastic Processes Representations of Deterministic Signals Finite-Energy Second-Order Stochastic Processes Second-Order Complex Stochastic Processes Complex Representations of Finite-Energy Second-Order Stochastic Processes Finite-Power Stochastic Processes Complex Wide-Sense-Stationary Processes Complex Representations of Real Wide-Sense-Stationary Signals 60.3 The Multivariate Complex Gaussian Density Function 60.4 Related Distributions Complex Chi-Squared Distribution Complex F Distribution Complex Beta Distribution Complex Student-t Distribution 60.5 Conclusion References 60.1 Introduction Much of modern digital signal processing is concerned with the extraction of information from signals which are noisy or which behave randomly while still revealing some attribute or parameter of a system or environment under observation. The term in popular use now for this kind of computation is statistical signal processing and much of this Handbook is devoted to this very subject. Statistical signal processing is classical statistical inference applied to problems of interest to electrical engineers with the added twist that answers are often required in real time perhaps seconds or less. Thus computational algorithms are often studied hand-in-hand with statistics. One thing that separates the phenomena electrical engineers study from that of agronomists economists or biologists is that the data they process are very often complex that is the data points come in pairs of the form x C jy where x is called the real part y the imaginary part and j V_1. Complex numbers are entirely a human intellectual creation .